| We make a detailed study of the numerical ranges of completely nonunitary contractions T with the property rank =1 on a finite-dimensional Hilbert space. We show that such operators are completely characterized by the Poncelet property of their numerical ranges, namely, an n-dimensional contraction T is in the above class if and only if for any point on the unit circle there is an (n+1)-gon which is inscribed in the unit circle, circumscribed about W(T) and has as a vertex. We also obtain some applications of their numerical range. |